Sums of four polygonal numbers with coefficients

Xiang-Zi Meng, Zhi-Wei Sun

Published 2016-08-08Version 1

Let $m\ge3$ be a positive integer. The polygonal numbers of order $m+2$ are given by $p_{m+2}(n)=m\binom n2+n$ $(n=0,1,2,\ldots)$. For $(a,b)=(1,1),(2,2),(1,3),(2,4)$, we study whether any sufficiently large integer can be expressed as $$p_{m+2}(x_1) + p_{m+2}(x_2) + ap_{m+2}(x_3) + bp_{m+2}(x_4)$$ with $x_1,x_2,x_3,x_4$ nonnegative integers. We show that the answer is positive if $(a,b)\in\{(1,3),(2,4)\}$, or $(a,b)=(1,1)\ \&\ 4\mid m$, or $(a,b)=(2,2)\ \&\ m\not\equiv 2\pmod 4$. In particular, we confirm a conjecture of Z.-W. Sun which states that any natural number can be written as $p_6(x_1) + p_6(x_2) + 2p_6(x_3) + 4p_6(x_4)$ with $x_1,x_2,x_3,x_4$ nonnegative integers.